Preheat the oven to 375°F. Spray a donut pan with nonstick baking spray with flour.
Beat the butter and sugar until creamy. Add the yogurt, egg, and vanilla and beat again.
Stir together the dry ingredients. Add them alternately with the buttermilk to the butter mixture.
Stir in the shredded apples gently. Spoon the batter into a large Ziploc bag and cut on tip off. Pipe the batter into the donut pan, filling each cavity halfway.
Bake for 12 minutes, then let the donuts cool in the pan for 2 minutes before gently removing them and placing on a wire rack.
Wipe the donut pan out with a paper towel and repeat the steps with the rest of the donut batter.
Place the butter, brown sugar, and milk in a saucepan. Bring to a boil and simmer 1 minute. Remove from the heat and stir in the vanilla and powdered sugar.
Dip the tops of the donuts in the glaze and sprinkle with pecan chips. Store in a loosely sealed container.